The global behaviors for defocusing wave equations in two dimensional exterior region

We study the defocusing semilinear wave equation in R×R2\K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}\times {\mathbb {R}}^2\backslash {{\mathcal {K}}}$$\end{document} with the Dirichlet boundary condition, where K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {K}}}$$\end{document} is a star-shaped obstacle with smooth boundary. We first show that the potential energy of the solution will decay appropriately. Based on it, we show that the solution also pointwisely decays to 0. Finally, we show that the solution scatters both in energy space and the critical Sobolev space. In general, we show that most of the conclusions obtained in [20], which hold on R1+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{1+2}$$\end{document}, remain valid on R×R2\K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}\times {\mathbb {R}}^{2}\backslash {{\mathcal {K}}}$$\end{document}.